Further Exploration of Distance Functions

Objective

SWBAT graph a piecewise function or an absolute value function from a verbal description of a distance vs. time relationship

Big Idea

Are there function rules to fit these relationships? Students apply prior knowledge to a new problem and find functions to fit their data. The optional use of technology allows them to check models on their own.

Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.

Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.

Warm-Up

30 minutes

Giving students time to understand and solve problems helps them develop deeper understanding of the concepts required to solve the problems. It also gives them the chance to really think about what they are doing, rather than just rush through the work or try to memorize procedures. Today, I give students the same types of problems as yesterday to enable them to deepen their understanding. In doing so I want to leverage repetition, as well as time, to increase my students' learning.

I continue to ask students questions about their choice of level, especially on the first section of the Warm-up. The Level A problems get easy quickly for most students, so I am persistent in pushing students to try more challenging problems.

In the second section on distance functions, I like asking the open question: "How could you write an equation to fit this graph?" It is almost always true that in each class, there are some students who think of piecewise functions more intuitively, while others find absolute value functions to be more useful. This makes it really easy to get a conversation started between students who used different approaches (sometimes you have to move students around to make this happen) by asking them to talk about whether both types of equations work to fit the data (MP3). This is also a good chance to allow them to check their answers using graphing technology (MP5) without telling them how to do so.

Today, I start asking students questions like:

How do you choose inputs to the function that enable you to see the whole graph of the function?

Is it possible to draw the graph without even making the data table?

Is it possible to make the function rule without the graph and data table?

These types of questions get them thinking about the next steps and to start looking for shortcuts and making generalizations (MP7 & MP8). Even if they can't answer these questions today, they will start thinking this way.

Investigation

Closing

The closing of this lesson is a great opportunity to discuss the benefits of multiple representations with students. To start the conversation, I write each of these phrases on the board:

Data table

Verbal description

Absolute Value Function

Piecewise Function

Graph

Description of graph transformations

I arrange them in a sort of web, with no arrows or connections drawn, and I ask students to think about how all these representations fit together. I ask to draw their own arrows between the connections they can make. For instance, if they can turn the verbal description into a data table, I ask them to draw an arrow between these two terms in this direction. We will be using this same web of terms to help students keep track of their deepening understanding of the learning target. After students draw in some arrows that make sense to them, I ask them to write a brief note about what they do and don't understand so far. I collect this organizer from them, because I will pass it back to them each day to add to it.

This kind of task enables students to see how they are building knowledge from day to day, and helps them hold themselves accountable to making progress each day.